You can't keep ice fresh. Fresh ice is freshly made. If it's kept, it's not fresh. Even if it's kept frozen.
To the extent that you're going to keep ice at all, if you don't like plastic, use a stainless, glass or ceramic container, and cover it with cling wrap if the container doesn't have an air-tight lid. Keeping it covered helps prevent the ice from absorbing freezer aromas, and those aromas are a big part of why ice stales.
If you decide to use a stainless (i.e., "inox") tray, cover it with cling wrap after filling but before freezing.
You might want to look for an ice bucket which uses a stainless insert. Someone must make them.
Given that you are very picky, don't need ice very often, but need a fair bit when you do, you might want to consider a portable ice maker. They're small enough to store, get the first dozen cubes out in around 15 minutes, make 25# or so pounds a day, allow you to choose your cube size, and at about $100 are fairly affordable.
If it's any consolation, I don't have one.